From 68d1ca6d03c70d827ab5e963e66d217f3565fad6 Mon Sep 17 00:00:00 2001 From: jgarnek Date: Sun, 22 Aug 2021 21:48:21 +0200 Subject: [PATCH] zapis w bazie dziala w polowie --- superelliptic.ipynb | 260 ++++++++++++++++++++++++++++++++++++++------ 1 file changed, 229 insertions(+), 31 deletions(-) diff --git a/superelliptic.ipynb b/superelliptic.ipynb index 8a8f046..7f505d1 100644 --- a/superelliptic.ipynb +++ b/superelliptic.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 19, + "execution_count": 115, "metadata": {}, "outputs": [], "source": [ @@ -35,8 +35,8 @@ " Fxy = FractionField(Rxy)\n", " \n", " basis = {}\n", + " k = 0\n", " if j == 'all':\n", - " k = 0\n", " for j in range(1, m):\n", " for i in range(1, r):\n", " if (r*j - m*i >= delta):\n", @@ -44,14 +44,13 @@ " k = k+1\n", " return basis\n", " else:\n", - " k = 0\n", " for i in range(1, r):\n", " if (r*j - m*i >= delta):\n", " basis[k] = superelliptic_form(self, Fxy(x^(i-1)/y^j))\n", " k = k+1\n", " return basis\n", " \n", - " def basis_de_rham(self, j = 'all'):\n", + " def degree_and_basis_de_rham(self, j = 'all'):\n", " f = self.polynomial\n", " m = self.exponent\n", " p = self.characteristic\n", @@ -61,21 +60,39 @@ " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " basis = {}\n", + " degrees0 = {}\n", + " degrees1 = {}\n", + " t = 0\n", " if j == 'all':\n", " for j in range(1, m):\n", " holo = C.basis_holomorphic_differentials(j)\n", " for k in range(0, len(holo)):\n", - " basis[k] = superelliptic_cech(self, holo[k], superelliptic_function(self, Rx(0))) \n", - " k = len(basis)\n", - " \n", + " basis[t] = superelliptic_cech(self, holo[k], superelliptic_function(self, Rx(0))) \n", + " g = Rx(holo[k].jth_component(j))\n", + " degrees0[t] = (g.degree(), j)\n", + " t += 1\n", " for i in range(1, r):\n", " if (r*(m-j) - m*i >= delta):\n", " s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f\n", " psi = Rx(cut(s, i))\n", - " basis[k] = superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^j/x^i)))\n", - " k = k+1\n", - " return basis\n", - " \n", + " basis[t] = superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^j/x^i)))\n", + " degrees0[t] = (psi.degree(), j)\n", + " degrees1[t] = (-i, j)\n", + " t += 1\n", + " return basis, degrees0, degrees1\n", + " \n", + " def degree_de_rham(self, i, j='all'):\n", + " basis, degrees0, degrees1 = self.degree_and_basis_de_rham(j)\n", + " if i==0:\n", + " return degrees0\n", + " \n", + " if i==1:\n", + " return degrees1\n", + " \n", + " def basis_de_rham(self, j = 'all'): \n", + " basis, degrees0, degrees1 = self.degree_and_basis_de_rham(j)\n", + " return basis\n", + " \n", "def reduction(C, g):\n", " p = C.characteristic\n", " Rxy. = PolynomialRing(GF(p), 2)\n", @@ -110,7 +127,7 @@ " m = C.exponent\n", " g = reduction(C, g)\n", "\n", - " g1 = RR(0)\n", + " g1 = Rxy(0)\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", @@ -119,7 +136,7 @@ " for j in range(0, m):\n", " if j==0:\n", " G = coff(g, 0)\n", - " g1 += G\n", + " g1 += FxRy(G)\n", " else:\n", " G = coff(g, j)\n", " g1 += Fxy(y^(j-m)*f*G)\n", @@ -143,9 +160,13 @@ " \n", " def jth_component(self, j):\n", " g = self.function\n", - " Rxy. = PolynomialRing(GF(p), 2)\n", - " g = Rxy(g)\n", - " return g.coefficient(y^j)\n", + " C = self.curve\n", + " p = C.characteristic\n", + " Rx. = PolynomialRing(GF(p))\n", + " Fx. = FractionField(Rx)\n", + " FxRy. = PolynomialRing(Fx)\n", + " g = FxRy(g)\n", + " return coff(g, j)\n", " \n", " def __add__(self, other):\n", " C = self.curve\n", @@ -219,11 +240,13 @@ " \n", " def jth_component(self, j):\n", " g = self.form\n", + " C = self.curve\n", + " p = C.characteristic\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", " Fxy = FractionField(FxRy)\n", - " Ryinv = PolynomialRing(Fx)\n", + " Ryinv. = PolynomialRing(Fx)\n", " g = Fxy(g)\n", " g = g(y = 1/y_inv)\n", " g = Ryinv(g)\n", @@ -272,8 +295,75 @@ " C = self.curve\n", " return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)\n", " \n", + " def mult(self, constant):\n", + " C = self.curve\n", + " w1 = self.omega0.form\n", + " f1 = self.f.function\n", + " w2 = superelliptic_form(C, constant*w1)\n", + " f2 = superelliptic_function(C, constant*f1)\n", + " return superelliptic_cech(C, w2, f2)\n", + " \n", " def __repr__(self):\n", " return \"(\" + str(self.omega0) + \", \" + str(self.f) + \", \" + str(self.omega8) + \")\" \n", + " \n", + " def basis_coeffs(self):\n", + " C = self.curve\n", + " g = self.f\n", + " basis = C.basis_de_rham()\n", + "\n", + " def coordinates(self):\n", + " print(self)\n", + " C = self.curve\n", + " p = C.characteristic\n", + " m = C.exponent\n", + " Rx. = PolynomialRing(GF(p))\n", + " Fx = FractionField(Rx)\n", + " FxRy. = PolynomialRing(Fx)\n", + " g = C.genus()\n", + " degrees0 = C.degree_de_rham(0)\n", + " degrees0_inv = {b:a for a, b in degrees0.items()} \n", + " degrees1 = C.degree_de_rham(1)\n", + " degrees1_inv = {b:a for a, b in degrees1.items()}\n", + " basis = C.basis_de_rham()\n", + " \n", + " omega = self.omega0\n", + " fct = self.f\n", + " \n", + " if fct.function == Rx(0) and omega.form == Rx(0):\n", + " for j in range(1, m):\n", + " omega_j = Fx(omega.jth_component(j))\n", + " if omega_j != Fx(0):\n", + " d = d = degree_of_rational_fctn(omega_j)\n", + " index = degrees0_inv[(d, j)]\n", + " a = coeff_of_rational_fctn(omega_j)\n", + " a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j))\n", + " elt = self - basis[index].mult(a/a1)\n", + " return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, 2*g)])\n", + " \n", + " for j in range(1, m):\n", + " fct_j = Fx(fct.jth_component(j))\n", + " if (fct_j != Rx(0)):\n", + " d = degree_of_rational_fctn(fct_j)\n", + " \n", + " if (d, j) in degrees1.values():\n", + " index = degrees1_inv[(d, j)]\n", + " a = coeff_of_rational_fctn(fct_j)\n", + " a1 = coeff_of_rational_fctn(basis[index].f.jth_component(j))\n", + " elt = self - basis[index].mult(a/a1)\n", + " return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, 2*g)])\n", + " \n", + " if d<0:\n", + " a = coeff_of_rational_fctn(fct_j)\n", + " elt =- superelliptic_cech(C, elt.omega0, elt.f - FxRy(y^j*x^d)).mult(a)\n", + " return elt.coordinates()\n", + " \n", + " if (fct_j != Rx(0)):\n", + " G = superelliptic_function(C, y^j*x^d)\n", + " a = coeff_of_rational_fctn(fct_j)\n", + " elt =-superelliptic_cech(diffn(G), G).mult(a)\n", + " return elt.coordinates()\n", + "\n", + " return vector(2*g*[0])\n", " \n", "def degree_of_rational_fctn(f):\n", " Rx. = PolynomialRing(GF(p))\n", @@ -285,6 +375,18 @@ " d2 = f2.degree()\n", " return(d1 - d2)\n", "\n", + "def coeff_of_rational_fctn(f):\n", + " Rx. = PolynomialRing(GF(p))\n", + " Fx = FractionField(Rx)\n", + " f = Fx(f)\n", + " f1 = f.numerator()\n", + " f2 = f.denominator()\n", + " d1 = f1.degree()\n", + " d2 = f2.degree()\n", + " a1 = f1.coefficients(sparse = false)[d1]\n", + " a2 = f2.coefficients(sparse = false)[d2]\n", + " return(a1/a2)\n", + "\n", "def coff(f, d):\n", " lista = f.coefficients(sparse = false)\n", " if len(lista) <= d:\n", @@ -299,30 +401,86 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 116, "metadata": {}, "outputs": [ { - "ename": "TypeError", - "evalue": "unsupported operand parent(s) for +: 'Real Field with 53 bits of precision' and 'Fraction Field of Univariate Polynomial Ring in x over Finite Field of size 5'", + "data": { + "text/plain": [ + "{0: ((x/y) dx, 2/x*y, ((x^3*y^5 - x^3 + x - 1)/(x^2*y^6)) dx),\n", + " 1: (((-1)/y) dx, 2/x^2*y, ((-x^3*y^5 + x^3 - 2*x - 2)/(x^3*y^6)) dx),\n", + " 2: (((-2*x)/y^2) dx, 2/x*y^2, ((-2*x^3*y^3 + x^3 - 1)/(x^2*y^5)) dx),\n", + " 3: ((1/y^2) dx, 2/x^2*y^2, ((x^3*y^3 - 2*x^3 + 2*x - 2)/(x^3*y^5)) dx),\n", + " 4: ((1/y^3) dx, 0, (1/y^3) dx),\n", + " 5: (0 dx, 2/x*y^3, ((-2*x^3 - x - 1)/(x^2*y^4)) dx),\n", + " 6: ((1/y^4) dx, 0, (1/y^4) dx),\n", + " 7: ((2*x/y^4) dx, 2/x*y^4, ((2*x^3 - 2*x*y - y)/(x^2*y^4)) dx),\n", + " 8: ((1/y^5) dx, 0, (1/y^5) dx),\n", + " 9: ((x/y^5) dx, 0, (x/y^5) dx),\n", + " 10: ((1/y^6) dx, 0, (1/y^6) dx),\n", + " 11: ((x/y^6) dx, 0, (x/y^6) dx)}" + ] + }, + "execution_count": 116, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "p = 5\n", + "C = superelliptic(x^3 + x + 2, 7, p)\n", + "C.basis_de_rham()\n", + "#C.basis_holomorphic_differentials()" + ] + }, + { + "cell_type": "code", + "execution_count": 113, + "metadata": {}, + "outputs": [], + "source": [ + "RxRy. = PolynomialRing(GF(p), 2)\n", + "w1 = superelliptic_cech(C, superelliptic_form(C, (1/y^5)), superelliptic_function(C, 0))\n", + "w2 = superelliptic_cech(C, superelliptic_form(C,2*x/y^4), superelliptic_function(C, 2/x*y^4))\n", + "w = w1+w2+w2" + ] + }, + { + "cell_type": "code", + "execution_count": 114, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(((-x*y + 1)/y^5) dx, 4/x*y^4, ((-x^3*y + x*y^2 + x^2 - 2*y^2)/(x^2*y^5)) dx)\n" + ] + }, + { + "ename": "AttributeError", + "evalue": "'superelliptic_function' object has no attribute 'jth_coordinate'", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuperelliptic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m7\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbasis_de_rham\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0;31m#C.basis_holomorphic_differentials()\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - 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"\u001b[0;31mTypeError\u001b[0m: unsupported operand parent(s) for +: 'Real Field with 53 bits of precision' and 'Fraction Field of Univariate Polynomial Ring in x over Finite Field of size 5'" + "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mw\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", + "\u001b[0;32m\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 341\u001b[0m \u001b[0mindex\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdegrees1_inv\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 342\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 343\u001b[0;31m \u001b[0ma1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbasis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mindex\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mjth_coordinate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 344\u001b[0m \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mbasis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mindex\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0ma1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 345\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0ma1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mvector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mGF\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mindex\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mAttributeError\u001b[0m: 'superelliptic_function' object has no attribute 'jth_coordinate'" ] } ], "source": [ - "C = superelliptic(x^3 + x + 2, 7, 5)\n", - "C.basis_de_rham()\n", - "#C.basis_holomorphic_differentials()" + "w.coordinates()" + ] + }, + { + "cell_type": "code", + "execution_count": 65, + "metadata": {}, + "outputs": [], + "source": [ + "a, b, c = C.degree_and_basis_de_rham()" ] }, { @@ -653,6 +811,46 @@ "A.derivative(x)" ] }, + { + "cell_type": "code", + "execution_count": 42, + "metadata": {}, + "outputs": [], + "source": [ + "dict1 = {}\n", + "dict1[3] = 5\n", + "dict1[6] = 121" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": {}, + "outputs": [], + "source": [ + "degrees1_inv = {b:a for a, b in dict1.items()}" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "{5: 3, 121: 6}" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "degrees1_inv" + ] + }, { "cell_type": "code", "execution_count": null,